are 202, spring 2018 welfare: tools and applications ...fally/courses/are202lecture2.pdf · income...

ARE 202, Spring 2018

Welfare: Tools and Applications

Thibault Fally

Lecture notes 02 – Price and Income Effects

ARE202 - Lec 02 - Price and Income Effects 1 / 74

Plan

1. Preferences and utility

• Preferences and utility, Debreu’s theorem

• Marshallian demand

• Examples of utility functions and demand

2. About aggregation and RUM

3. Duality

• Hicksian Demand

• Shephard’s Lemma and Roy’s Identity

• Giffen goods: example from Jensen and Miller (2008)


1) Preferences, Utility and Demand

Preferences and utility

Marshallian demand

Demand and price elasticities

Illustrating income effects

Examples of utility functions


Some definitions

Rational preferences: Preferences % on X are rational if:

• Completeness: For all x , y ∈ X , we have x % y and/or y % x

• Transitivity: For all x , y , z ∈ X , x % y and y % z implies x % z

Other definitions:

• Preferences % on X are monotone if x ≫ y implies x ≻ y , and strictlymonotone if x ≥ y and x 6= y implies x ≻ y

• Preferences % on X are continuous if for all xn, yn such that xn % yn,xn → x and yn → y , then x % y .

• Preferences % on X are locally non-satiated if for every x ∈ X andε > 0, there is a y ∈ X such that ‖x − y‖ < ε and x ≻ y

• Preferences % on X are convex if for every α ∈ (0, 1), y % x andz % x then αy + (1−α)z % x (≻ if strictly convex)

• Preferences are homothetic if for any α > 0, x ∼ y implies αx ∼ αy


Utility representation:

Utility function such that: x % y ⇔ U(x) ≥ U(y)

Debreu’s theorem:Let X ⊂ Rn. Preferences % on X have a continuous utility representationif and only if these preferences are (check needed conditions):

rational?

monotone?

strictly monotone?

continuous?

locally non-satiated?

convex?

strictly convex?

homothetic?

Counter-example: preferences that don’t have a continuous rep’?


Homothetic preferences:

Preferences such that, for any α > 0, x ∼ y implies αx ∼ αy

Proposition:Any homothetic, continuous and monotonique preference relation can berepresented by a utility function that is homogeneous of degree one.


Utility maximization problem:

max u(x)

such that: p.x ≤ w

Proposition:It has a unique solution x(p,w) (Marshallian or Walrasian demand) if (checkwhat is needed):

u(x) is continuous?

u(x) is strictly quasi-concave?

preferences are homothetic?

corresponding preferences are locally non-satiated?

Notes:

Example of preferences that we will use but does not satisfy all theseconditions: Leontief preferences

p.x ≤ w is the budget constraint (a.k.a. Walrasian set)


Properties of Marshallian demand

We have:

x(w , p) is homogeneous of degree zero

Moreover, if preferences are locally non-satiated:

p.x(w , p) = w

In general, we will also assume that u(x) is differentiable as many times asneeded.


Indirect utility and marginal utility of wealth

Indirect utility: Utility associated with the chosen bundle x(p,w):

V (p,w) = U(x(p,w)) = maxU(x) such that p.x ≤ w

Marginal utility of wealth:

∂V∂w =

∑

i∂U∂xi.∂xi∂w =


More definitions

Def 1: Price elasticity: εPi = ∂ log xi∂ log pi

• εPi < 0 Giffen good

• εPi > 1 Price-elastic good

Def 3: Income elasticity: εIi =∂ log xi∂ logw

• εIi < 0 Inferior good

• εIi ∈ [0, 1] Normal good

• εIi > 1 Luxury good

Def 3: Elasticity of substitution: σji =d log(xj/xi )d log(Ui/Uj )

=d log(xj/xi )d log(pi/pj )

(curvature of indifference curve)


Illustrating wealth effects

Engel curves

Consumption against income

Wealth expansion pathsOptimal consumption baskets as income varies(holding prices constant)


Wealth expansion paths with homothetic preferences


Engel curves

Definition: Consumption against income

Left: income-inelastic good; Right: luxurious good

w w

x x


Engel aggregation

The weighted average of income elasticities has to equal unity:

∑

i pixiεInci

∑

i pixi= 1

There can’t be only inferior goods or only luxury goods

Complementarity

Goods i and j are “gross substitutes” if ∂xi∂pj

> 0 (e.g. gas and cars),

and “gross complements” otherwise (e.g. different brands of a good).

Note: better definition (“net substitutes”): using Hicksian demand


Examples to know well (1/5):

Cobb-Douglas: u(x) =∑

i αi log xi with∑

i αi = 1





i αi = 1

We get: xi (p,w) = αiwpi

Elasticities:

Income elasticity = 1 for all goods

Own price elasticity = 1 for all goods, cross price elasticity = 0

Elasticity of substitution = 1



Stone-Geary: u(x) =∑

i αi log(xi − φi ) with φi > 0

with choke price if φi < 0



u(x) =∑

i αi log(xi − φi )

We get: xi (p,w) =αi (w−

∑j pjφj )

pi+ φi if αiw

pi> −φi

Elasticities:

w −∑

j pjφj = “disposable income”

Income and price elasticities depend on relative size of φi ’s

Example: with φi = −φ < 0, only low-price commodities are consumed.Consumption of higher-price commodities positive only above a certainthreshold of wealth (see problem set 2).



Leontief: u(x) = mini xi/αi

Linear: u(x) =∑

i αixi



Leontief: u(x) = mini xi/αi

⇒ xi =αiw∑j αjpj

, and therefore xixj= αi

αj

Linear: u(x) =∑

i αixi

⇒ xi =wpi

if piαi

= minj

pjαj

, xi = 0 otherwise



CES: u(x) =[∑

i xρi

]1ρ

Limit cases when ρ = 0? ρ = 1?



CES: u(x) =[∑

i xρi

]1ρ

we get: xi =(

piP

)

−σ wP,

with σ = 11−ρ

and price index P =[∑

i p1−σ

i

]1

1−σ

Income elasticity = 1

Own price elasticity = σ − (σ − 1)si (si share of i in expenditures)

Cross price elasticity = − (σ − 1)sj (w.r.t. price of good j)

Own (resp. cross) elasticity equals ≈ σ (resp. 0) if market share is small

Elasticity of substitution = σ



Separable and quasi-linear: u(x) = x0 +∑

i ui (xi )



Separable and quasi-linear: u(x) = x0 +∑

i ui (xi )

“Numeraire” x0: we get: λ = p0 for the numeraire

Practical to normalize p0 = 1 ⇒ Lagrange multiplier λ equals one

u′i (xi ) = pi yields demand: xi = Di (pi ) that only depends on price pi

In turn, consumptino of numeraire is x0 = w −∑

j pjDj(pj)

No income effect (income elast. = 0)

except for numeraire (income elast. > 1)


Continuous versions and combinations

Integrating over goods i , often indexed by i ∈ [0, 1]:

Cobb-Douglas: u(x) =∫

iαi log xi di with

∫

iαi di = 1

Stone-Geary: u(x) =∫

ilog(xi − φi ) di

Linear: u(x) =∫

iαixi di

CES: u(x) =[∫

ixρi di

]1ρ

Separable and quasi-linear: u(x) = x0 +∫

iui (xi ) di

and combinations, e.g.:

u(x) =∑

k αk log[∫

ixρik di

]1ρ


Other examples (1/4):

CRIE: u(x) =∑

i x

σi−1

σi

i (see Caron, Fally and Markusen 2014)

Advantage: constant ratio of income elasticity σi

σjacross two goods


Pigou’s law

With separable utility u(x) =∫

iui (xi ) di , the price elasticity is propor-

tional to the income elasticity:

∂ log xi∂ logw

=∂ log xi∂ log pi

∂ log λ

∂ logw

when good i has a negligible market share. Hence:

∂ log xi∂ logw

∂ log xj∂ logw

=

∂ log xi∂ log pi∂ log xj∂ log pj

This is a strong restrictionThis feature is often rejected in the data (Deaton 1974)

Implicit utility functions as above can address this issuesee Comin, Lashkari and Mestieri (2017)



Gorman implicit utility: 1 =∑

i

(

xigi (u)

)ρ(Comin et al. 2017)

Advantage: Non-separable, no link bw income and price elasticities(Note: one can actually have ρ depend on u, see Fally 2017)



Quality w outside good: u(x) = U(uG (x , z), z) (Faber and Fally 2017)

with uG (x , z) =[∫

i(ϕi (z)xi )

ρ(z) di]

1ρ(z)

Advantage: Price elasticity σ(z) and Quality valuation ϕi (z) of goodi vary with consumption of outside good z and therefore income.


Other examples: AIDS (Deaton Mullbauer)“Almost Ideal Demand System” (acronym chosen in the 70’s)

Assume: log e(p, u) = a(p) + ub(p)

We get market shares: xiw= Ai (p) + Bi (p) logw

Further imposing:

a(p) = α0 +∑

j

αj log pj +1

2

∑

j

∑

k

γjk log pj log pk

b(p) = β0∏

k

pβk

k

with 0 = 1−∑

j

αj =∑

j

βj =∑

j

γjk and γkj = γjk

we get: xiw= αi +

∑

j γij log pj + βi log(w/P)with logP ≈

∑

ixiwlog pj .


Demand with single aggregator

Gorman (1972), Fally (2017)

If demand depends on own income w , own price pi and a common priceaggregator Λ, then it must take one of these two forms:

qi (w , pi ,Λ) = Di (F (Λ)pi/w)/H(Λ)

or:qi (w , pi ,Λ) = Gi (Λ)(pi/w)−σ(Λ)

Conversely, these demand systems are integrable if:

εDiεF < εH for all pi/w and ΛGi (Λ) increases sufficiently fast with Λ (see Fally 2017 for conditions)


Notes on the choice of preferences mentioned earlier

Cobb-Douglas: Used across broad categories of goods when we want to hold con-stant expenditures shares for simplicity and tractability.

CES: workhorse in Macro, Trade, etc. as it is homothetic and works very well withmonopolistic competition.

Stone-Geary: Most simple way to get non-homotheticity, but income effects con-verge very quickly for higher levels of income, substitution effects are too restrictive.

AIDS: Very-widely used even today. Flexible income effects and price effects, butnot very tractable. An advantage to Stone Geary is that expenditure shares dependon log income. Problems with bounds (corner solutions for expenditure shares)

Fieler (2011), Ligon (2016), Caron et al (2014), etc.: Behave almost like AIDSw.r.t income (log expenditures vary with log income), but subject to Pigou’s law

Comin et al (2016): Behave almost like AIDS w.r.t income (log expenditures varywith log income), simple price effects as in CES, yet avoids Pigou’s curse.

Faber and Fally (2017): very amenable to empirical estimation, do not impose howincome affect substitution and price effects, flexible income effect through quality.

Kimball (1995): Homothetic with very flexible own-price elasticities (PS1 part B).


Plan


2. Aggregation

3. Duality


2) Aggregation

When can we express aggregate demand as a function of pricesand aggregate wealth?

Discrete-choice models


Gorman’s Aggregation Theorem

When can we express aggregate demand as a function of prices andaggregate wealth, irrespective of the distribution of wealth?

Answer:When one can express indirect utility with a Gorman form:

vh(p,wh) = ah(p) + b(p)wh

Note: Weaker restrictions can be imposed if we specify the distributionof wealth.

Examples: quasi-linear preferences, identical Stone-Geary preferences,identical homothetic pref. (where vi (p,wi ) = b(p)wi )


Implication for wealth expansion paths

With Gorman form, Marshallian demand is linear in wealthThis can be shown easily using Roy’s identity:

xhi = −

∂vh(p,wh)∂pi

∂vh(p,wh)∂wh

= −1

b(p).∂vh(p,wh)

∂pi

This implies linear wealth expansion paths with Gorman

linear Engel curves

and expenditure functions linear in u

Gorman form: sometimes called “quasi-homothetic”


Wealth expansion paths with Gorman preferences:



or “Random Utility Models”, pioneered by McFadden

Each consumer only buys one good (within a category)Focus on a specific industry and often assume quasi-linear pref.

Individuals may differ in their taste for attributes of goodsand have idiosyncratic taste shock for each good

Typically, indirect utility of consumer z with choice i is:

Uzi = αz(wz − pi ) + φz(Zi ) + ǫzi

hence income effects drop out (quasi-linear preferences)

See Berry, Levinsohn and Pakes (1995), aka BLP, for estimationCore topic in IO and Environmental Econ courses



We can also mix discrete choice (each consumer buys one brand) withcontinuous quantities (how much it buys from that brand).

Discrete but continuous quantity choice with type-II extreme value dis-tribution for εzi leads exactly to CES on aggregate

Uz = maxi∈Ω, qzi

[log qzi + logϕi + µǫzi ]

... equivalent to:

U = log

[

∑

i∈Ω

(qi logϕi )σ−1σ

]

after aggregating across individuals (Anderson, de Palma, Thisse 1987)with elasticity of substitution σ = 1 + 1

µ


Plan


2. Aggregation

3. Duality


3) Price effects and duality:

• Why do we need other tools?Problems with indirect utility and Marshallian demand

• Definitions:

Dual problem

Hicksian Demand

Expenditure function

• Properties:

Shephard’s lemma

Slutsky equation

• Application: Are there Giffen Goods?Jensen and Miller (2008)


Indirect utility

Indirect utility: Utility associated with the chosen bundle x(p,w):

V (p,w) = U(x(p,w)) = maxU(x) such that p.x ≤ w

Issues with V ?

• We can redefine utility up to any increasing function f (U(x)) whichwould yield f (V (p,w)) for indirect utility.

⇒ Then how to interpret V if any other f (V (x)) would also work?

• How to compare individuals?

• How to put a dollar value on V ?


Price effects with Marshallian demand

Price effect: Why is the sign of ∂xi∂pi

not always positive?We were told that a demand curve is downward slopping...

This price effect depends on various things:

• Curvature of indifference curve

• Difference between indifference curves at different utility levels

⇒ Not so easy to illustrate / understand

Note: In comparison, wealth effects ∂xi∂w are easy to understand with Mar-

shallian demand: budget set shifts in or out by preserving relative prices andmarginal rate of substitution ∂U

∂x1/ ∂U∂x2

.


Price effect with Marshallian demand:


Example of positive price effect:


New tools needed

It is easier to capture movements along indifference curvesi.e. holding Utility fixed

This leads to a set of new tools:

• Expenditure function e(p, u): wealth required to get utility u withprices p (Note: e is concave in p)

• Hicksian demand hi (p, u): demand for good i as a function of utilityu and prices p

hi (p, u) = xi (p, e(p, u))

also called “compensated demand function”

For the story, Marshall was the first one to draw demand demand and supply curves.Hicks was the first one to carefully examine price effects.


Dual problems

Utility maximization problem

maxU(x)

such that: p.x = w

leads to Marshallian demand x(p,w) and indirect utility v(p,w)

Expenditures minimization problem

min p.x

such that: U(x) = u

leads to Hicksian demand h(p, u) and expenditure fctn e(p, u)


Understanding welfare and price effects

Using expenditure function to examine welfare: “Lecture notes 04”

(compensating variations and equivalent variations)

Today: focus on the price effect

First property:The price effect with the Hicksian demand is always negative:

∂hi (p, u)

∂pi< 0

as long as preferences are convex (i.e. utility quasi-concave)


Price effect with Hicksian demand:


Other properties

Lagrange multiplier in EMP inversely related to Lagrange multiplier inUMP:

λEMP =pi∂U∂xi

=1

λUMP

Note also that:

λEMP =∂e(p, u)

∂u

Moreover, the envelop theorem then gives:

∂e(p, u)

∂pi= hi (p, u)

This is called Shephard’s Lemma


Roy’s Identity

Equivalent of Shephard’s Lemma for Marshallian demand?

Exercise: Show that:

−

∂V (p,w)∂pi

∂V (p,w)∂w

= −−λUMP . xi (p,w)

λUMP= xi (p,w)

Applications: sometimes it is practical to specify indirect utility ratherthan demand and preferences. Roy’s identity then yields demand.

Example: Addilog:

V (p,w) =

∫

i

v(pi/w)di

CES is a special case with iso-elastic v(.), other cases are non-homothetic


Price effect

Linking Hicksian and Marshallian demand

Recall that hi (p, u) = xi (p, e(p, u)). Differentiating, we get:

∂hi (p,u)∂pj

= ∂xi (p,w)∂pj

+ ∂e(p,u)∂pj

. ∂xi (p,w)∂w

Rearranging: ∂xi (p,w)∂pj

= ∂hi (p,u)∂pj

− ∂e(p,u)∂pj

. ∂xi (p,w)∂w

Using Shephard’s Lemma, we obtain Slutsky Equation:

∂xi (p,w)

∂pj=

∂hi (p, u)

∂pj− hj .

∂xi (p,w)

∂w

Price effect = Substitution - Income effect

In elasticities: εMarshallij = εHicks

ij − sj . εMarshalliw


Summing up:


Price effects: three cases

Let’s come back to different types of goods depending on their wealth ef-fects:

Normal goods: Income effect reinforces the substitution effect

∂xi (p,w)∂pi

= ∂hi (p,u)∂pi

− hi .∂xi (p,w)

∂w < ∂hi (p,u)∂pi

< 0

Inferior goods: Income effect mitigate substitution effect

∂xi (p,w)∂pi

= ∂hi (p,u)∂pi

− hi .∂xi (p,w)

∂w > ∂hi (p,u)∂pi

(but still < 0)

Giffen goods: Income effect dominates the substitution effect

∂xi (p,w)∂pi

= ∂hi (p,u)∂pi

− hi .∂xi (p,w)

∂w > 0


Overall price effect:






Slope of demand


Slope of demand


Slope of demand


Examples

[see blackboard: expenditure function, Hicksian demand]

Using the utility functions examined previously:

Leontief and linear: u(x) = mini αixi and u(x) =∑

i αixi



i αi = 1

Stone-Geary: u(x) =∑

i log(xi − φi )

CES: u(x) =[∑

i xρi

]1ρ


Examples


Examples

⇒ Expenditure functions:

Leontief: e(u, p) = u.∑

i αipi

Linear: e(u, p) = u.minipi/αi

Cobb-Douglas: e(u, p) = u.∏

i

(

piαi

)αi

Stone-Geary: e(u, p) =∑

i piφi + u.∑

i

∏

i

(

piαi

)αi

CES: e(u, p) = u.[∑

i p1−σi

]1

1−σ


Giffen good: theoretical artifact?

Giffen behavior would require:

1 Very negative income elasticity:Staple for the Poor, substituted by other products by the Rich

2 Large consumption by the poor: the income effect in Slutsky equationis larger for larger consumption shares.

3 Low substitution ∂hi (p,u)∂pi

with other staples

Potatoes during the Great Irish Famine? (1845-52)


Jensen and Miller (AER 2008)

First study to carefully show evidence of Giffen behavior

Population:

about 1,300 households living with less than 1/day in Hunan andGansu provinces in 2005

Food items:

Hunan: rice; Gansu: wheat

Identification issues:

Demand shocks usually lead to positive correlations between prices andquantities when prices respond to changes in demand – this is not theproof that there are Giffen goods

Experimental setting:

Randomly give lower prices (rice or noodles) to some households during5 months (discounts worth about 10-25%)



They argue that they need the following conditions:

C1 Households are poor enough to face subsistence nutrition concerns

C2 Simple diet, including a basic and a fancy food

C3 a) This basic food constitutes a large part of the diet (e.g. rice)

b) Basic food is cheapest source of calories and has no ready substitute

C4 Households are not too poor either: they do not only consume thebasic good


Indifference curves




Staple price elasticities across households



Conclusions

Finally an example of Giffen good to provide in a micro class!!(other than potatoes during the great famine)

Applies to most common goods in the most populated country

Great identification strategy (not my role to discuss it here)

Great use of micro-theory

⇒ Giffen behavior seems to happen where theory would predict:Households that are poor but not starving, consuming a specific staplegood as main source of calories

I wish could more precisely disentangle price from wealth effects bycombining price discount with random cash transfers.


Exercise

Build a simple example of utility with two goods (e.g. rice and meat) thatgenerates Giffen goods as in Jensen and Miller (2008)?[Get inspiration from criteria C1 to C4]


Hamilton Method

Can we retrieve real income from consumption patterns? (Hamilton 2001)

Data: nominal income, approximation of relative prices

Goal: estimate an inflation bias µt common to all goods k .

Nakamura et al (2016) specify consumption shares as:

ωki ,t = ψk

i + βk log(Ci ,t/Pi ,t) + γk log(Pki ,t/Pi ,t) +

∑

x

ΘkxXi ,t + ǫi ,t

where Ci ,t denotes nominal expenditures at time t for individual(s) i .

Issues with this approach?


Missing inflation?

Each obs = income group / year


are 202, spring 2018 welfare: tools and applications ...fally/courses/are202lecture2.pdf · income...

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